Question: The arithmetic sequence $a_i$ is defined by the formula: $a_1 = 15$ $a_i = a_{i - 1} -7$ Find the sum of the first $660$ terms in the sequence.
Answer: Getting started Let's write out the first few terms of the series: $15 + 8 + 1 + (-6)...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $7$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {15})$ and the number of terms $(n = {660})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $660 -1= 659$ terms after the first term. The sequence decreases by $7$ for each new term. So, the sequence decreases by a total of $659 \cdot 7 = 4613$ from where it starts at $15$. That means the last term must be $15-4613 = {-4598}$. In other words, $a_n = {-4598}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{660}}&= \dfrac {\left({15} + ({-4598}) \right)}{2} \cdot {660} \\\\ S_{{660}} &= -2291.5 \left(660\right) \\\\ S_{{660}} &= -1{,}512{,}390\end{aligned}$ The answer $ -1{,}512{,}390 $